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In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, a central moment is a
moment
Moment or Moments may refer to:
* Present time
Music
* The Moments, American R&B vocal group Albums
* ''Moment'' (Dark Tranquillity album), 2020
* ''Moment'' (Speed album), 1998
* ''Moments'' (Darude album)
* ''Moments'' (Christine Guldbrand ...
of a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
of a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
about the random variable's
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
; that is, it is the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of a specified integer power of the deviation of the random variable from the mean. The various moments form one set of values by which the properties of a probability distribution can be usefully characterized. Central moments are used in preference to ordinary moments, computed in terms of deviations from the mean instead of from zero, because the higher-order central moments relate only to the spread and shape of the distribution, rather than also to its
location
In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ...
.
Sets of central moments can be defined for both univariate and multivariate distributions.
Univariate moments
The ''n''th
moment
Moment or Moments may refer to:
* Present time
Music
* The Moments, American R&B vocal group Albums
* ''Moment'' (Dark Tranquillity album), 2020
* ''Moment'' (Speed album), 1998
* ''Moments'' (Darude album)
* ''Moments'' (Christine Guldbrand ...
about the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
(or ''n''th central moment) of a real-valued
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
''X'' is the quantity ''μ''
''n'' := E
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''n'' where E is the expected value">expectation operator
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
. For a continuous probability distribution">continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
univariate
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
with
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
''f''(''x''), the ''n''th moment about the mean ''μ'' is
:
For random variables that have no mean, such as the Cauchy distribution, central moments are not defined.
The first few central moments have intuitive interpretations:
* The "zeroth" central moment ''μ''
0 is 1.
* The first central moment ''μ''
1 is 0 (not to be confused with the first
raw moment
In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total ma ...
or the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
''μ'').
* The second central moment ''μ''
2 is called the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
, and is usually denoted ''σ''
2, where σ represents the
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
.
* The third and fourth central moments are used to define the
standardized moment
In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invariant. ...
s which are used to define
skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimodal d ...
and
kurtosis
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosi ...
, respectively.
Properties
The ''n''th central moment is translation-invariant, i.e. for any random variable ''X'' and any constant ''c'', we have
:
For all ''n'', the ''n''th central moment is
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
of degree ''n'':
:
''Only'' for ''n'' such that n equals 1, 2, or 3 do we have an additivity property for random variables ''X'' and ''Y'' that are
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
:
:
provided ''n'' ∈ .
A related functional that shares the translation-invariance and homogeneity properties with the ''n''th central moment, but continues to have this additivity property even when ''n'' ≥ 4 is the ''n''th
cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will ha ...
κ
''n''(''X''). For ''n'' = 1, the ''n''th cumulant is just the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
; for ''n'' = either 2 or 3, the ''n''th cumulant is just the ''n''th central moment; for ''n'' ≥ 4, the ''n''th cumulant is an ''n''th-degree monic polynomial in the first ''n'' moments (about zero), and is also a (simpler) ''n''th-degree polynomial in the first ''n'' central moments.
Relation to moments about the origin
Sometimes it is convenient to convert moments about the origin to moments about the mean. The general equation for converting the ''n''th-order moment about the origin to the moment about the mean is
:
where ''μ'' is the mean of the distribution, and the moment about the origin is given by
:
For the cases ''n'' = 2, 3, 4 — which are of most interest because of the relations to
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
,
skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimodal d ...
, and
kurtosis
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosi ...
, respectively — this formula becomes (noting that
and
):
:
which is commonly referred to as
:
:
... and so on, following
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ot ...
, i.e.
:
because
The following sum is a stochastic variable having a ''compound distribution''
:
where the
are mutually independent random variables sharing the same common distribution and
a random integer variable independent of the
with its own distribution. The moments of
are obtained as
:
where
is defined as zero for
.
Symmetric distributions
In distributions that are
symmetric about their means (unaffected by being
reflected about the mean), all odd central moments equal zero whenever they exist, because in the formula for the ''n''th moment, each term involving a value of ''X'' less than the mean by a certain amount exactly cancels out the term involving a value of ''X'' greater than the mean by the same amount.
Multivariate moments
For a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
bivariate probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
with
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
''f''(''x'',''y'') the (''j'',''k'') moment about the mean ''μ'' = (''μ''
''X'', ''μ''
''Y'') is
:
Central moment of complex random variables
The ''n''th central moment for a complex random variable ''X'' is defined as
The absolute ''n''th central moment of ''X'' is defined as
The 2nd-order central moment ''β''
2 is called the
''variance'' of ''X'' whereas the 2nd-order central moment ''α''
2 is the
''pseudo-variance'' of ''X''.
See also
*
Standardized moment
In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invariant. ...
*
Image moment
In image processing, computer vision and related fields, an image moment is a certain particular weighted average ( moment) of the image pixels' intensities, or a function of such moments, usually chosen to have some attractive property or interp ...
*
*
Complex random variable In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. Complex random variables can alw ...
References
{{DEFAULTSORT:Central Moment
Statistical deviation and dispersion
Moment (mathematics)
fr:Moment (mathématiques)#Moment centré